In this explainer, we will learn how to use a graph to find the inverse of a function and analyze the graphs for the inverse of a function.
A relation, or mapping, transforms elements from one set onto elements from another. If every input in this mapping has exactly one output, it is called a function.
A function maps every element from an input set onto exactly one element from an output set. Functions can be either one-to-one (one input has one output) or many-to-one (many inputs map onto the same output).
If a function maps elements of the sets to , we can use the following notation:
The set of values that can be input into the function is called the domain, while the set of elements that come out is called the range.
If is a one-to-one function, it is said to be invertible. In other words, there exists an inverse to this function, defined such that the following definition holds.
Definition: Inverse Functions
Let be a function whose domain is the set and whose range is . is the inverse of with domain and range if
In other words, the inverse function “undoes” the original function. Take, for instance, the function . takes values of and multiplies them by 2. The inverse of is the function that “undoes” this process; hence, . Note that while there are algebraic processes to calculate the inverse of a function, it is outside the scope of this explainer to explore these in more detail.
By sketching the graph of and on the same set of axes, we can identify the single transformation that relates the graph of a function and its inverse.
The graph of is mapped onto the graph of by a single reflection in the line . This can be generalized for any invertible function .
Property: Graph of the Inverse
If is invertible, then the graph of is the same as the graph of the equation . This is obtained by reflecting the graph of across the line .
This is equivalent to switching the roles of and in the function, and so any coordinate for an inverse function can be found by switching the - and -coordinates of the corresponding point on the graph of the original function. For instance, the point with coordinates lies on the graph of , so the image of this point on the graph of has coordinates .
In our first example, we will demonstrate how to recognize the graph of an inverse function given the graph of the original function.
Example 1: Identifying the Graph of the Inverse of a Function
The following is the graph of .
Which one is the graph of the inverse function ?
Recall that, given a function , the graph of is obtained by reflecting the graph of in the line . This is equivalent to switching the - and -coordinates of each point that lies on the line .
Let’s begin by identifying any three points on the line . We will choose , , and .
To find the corresponding coordinates on the graph of the inverse of , we switch the - and -coordinates. The image of the points are therefore , , and respectively. Adding a straight line through these points, we obtain the graph of the inverse, . We can see from the following sketch that this is a reflection of the graph of across the dashed line .
The correct answer is (A).
In our previous example, we demonstrated that, by applying the definition of the inverse function, we can relate points on the graph of the function to the image of these points on the graph of the inverse. In our next example, we will perform a similar process on a cubic function.
Example 2: Relating the Graph of a Function to the Graph of Its Inverse Function
Shown is the graph of . Find the intersection of the inverse function with the -axis.
Recall that the graph of an inverse function is found by reflecting the graph of the original function across the line . In doing so, the roles of and are interchanged. This means that if the coordinate of the -intercept on the graph of is for some real , the image of this point on the graph of is . This is the value of an -intercept. So, to find the intersection of the inverse function with the -axis, we will find the coordinates of with the -axis and switch the roles of and .
The graph of passes through the -axis at . This means that the graph of passes through the -axis at .
Let’s demonstrate this graphically. The graph of is reflected in the dashed line as shown. The -intercept is 6, as expected.
In our previous example, we saw that if the coordinate of the -intercept on the graph of is for some real , the image of this point on the graph of is . It follows that the converse must also be true. Similarly, if the coordinate of the -intercept on the graph of is for some real , the image of this point on the graph of is .
In fact, we might even deduce more information about the domain and range of functions and their inverses. Since the outputs of are the inputs of , the range of is also the domain of . Similarly, since the inputs of are the outputs of , the domain of is the range of . Further, if a function has no inverse, it might be possible to restrict the domain of that function so that this new function does have an inverse.
For instance, consider the function whose graph is shown. This function fails the horizontal line test as shown below, so it is not one-to-one.
This means that if we reflect the graph of the function in the line , the resulting graph fails the vertical line test; it is a many-to-one mapping and is therefore not the graph of a function. Thus, does not have an inverse.
However, by restricting the domain of to , the function will pass the horizontal line test and will now be invertible. The graph of its inverse is found by reflecting across the line , such that the range of is the domain of , .
Property: Domain and Range of Inverse Functions
The range of a one-to-one function is the domain of the inverse function .
The domain of is the range of .
In our next example, we will demonstrate how to apply the relation between a graph of a function and the graph of its inverse to find their points of intersection.
Example 3: Using the Relationship between a Function and Its Inverse to Find Unknowns
The graphs of and its inverse intersect at three points, one of which is .
- Determine the value of .
- Find the -coordinate of the point marked on the figure.
- Find the -coordinate of the point marked on the figure.
Since the graph of passes through the point , we can substitute and into the equation to find the value of :
Solving for ,
The figure shows the graph of and its inverse. Since the value of the -intercept of is , the red plot is . This means the blue plot is . Hence, we can find the coordinates of point , which is the -intercept of the inverse function, by finding the coordinates of the -intercept of the original function and switching and . This is equivalent to reflecting this point across the line .
The -intercept of is . The -intercept of is therefore given by
Hence, the -coordinate of the point is .
Recall that the graph of is mapped onto by a reflection in the line . Since point is a point of intersection of and , it must lie on the line .
The point of intersection can therefore be found by solving the system of equations
Substituting in the first equation and rearranging gives
Since one of the points of intersection has a value , we know that is a factor of .
Dividing by gives us the factor . Hence, our previous equation can be written as
Finally, applying the quadratic formula to the equation , we find the solutions to be
Since the point of intersection lies in the first quadrant, its -coordinate must be positive.
Hence, the -coordinate of point is .
Let’s now demonstrate how to apply the properties of the graphs of inverse functions to sketch a function and its inverse.
Example 4: Finding a Graph of a Function That Is an Inverse of Itself
By sketching the graphs of the following functions, which is the inverse of itself?
To answer this question, we will sketch the graph of each function in turn, beginning with the graph of . This is a reciprocal function, with asymptotes at and .
Since finding the inverse is equivalent to switching the roles of and in the function, the asymptotes of the inverse of will be and . To sketch the graph of the inverse, we reflect the graph of the original function across the line .
This reflection maps the graph of onto itself, so the answer is A.
We will verify this by checking the other three functions, beginning with . is a many-to-one function and is therefore not invertible without performing some restrictions on its domain. Similarly, is a many-to-one function and has no inverse.
Next is the function . This is a cubic function that passes through the origin.
Reflecting the graph of the function in the line gives the following figure:
Since the graph of the function does not map onto the graph of its inverse, the answer cannot be C.
The correct answer is A, .
In our final example, we will demonstrate how restricting the domain of a function can make it invertible.
Example 5: Sketching Graphs of Functions to Determine Whether They Are Inverse
By sketching the graphs of and for , determine whether they are inverse functions.
The function is a many-to-one function—that is, a number of inputs to this function will have the same output. This means that it is not an invertible function. However, its domain has been restricted to , thereby creating a one-to-one function that does have an inverse.
Since the function is quadratic with a positive leading coefficient, it is a U-shaped parabola that passes through the origin. Restricting this to values of gives the following figure.
We will now sketch the graph of on the same axes. This is a transformation of the graph of by a horizontal stretch of scale factor 2. Its graph is shown below.
It does appear that these functions are inverses of one another, since it looks like they have been reflected across the line . We will check by looking at their point of intersection. If this lies on the line , its - and -coordinates will be equal.
To find this point, we will solve :
The solutions to this equation are or .
Substituting into either function gives . Since the - and -values are the same, we know the point of intersection of the curves lie on the line .
Similarly, substituting into either function gives . The point also lies on the line , as demonstrated in the following figure.
We can also check whether these functions are inverses of one another by looking at a couple of points on each line.
The point lies on the curve . The image of this point after a reflection across is . If this lies on the curve , substituting into this equation will give :
The image of the point after a reflection across lies on the curve . Hence, it appears that and for are inverses of one another.
Of course, even though we have looked at a handful of points, this is not quite enough. We could verify this by using graphing software to plot the graphs and their reflections across .
In our previous example, we demonstrated how to identify a series of points after a reflection across the line and use this information to determine whether a pair of functions were inverses of one another.
We could verify the result more stringently by using the definition of an inverse function; if be a function whose domain is the set and whose range is , is the inverse of with domain and range if
Evaluating the composite function , gives
They are inverse functions.
Let’s finish by recapping the key concepts from this explainer.
- If is a function whose domain is the set and whose range is , is the inverse of with domain and range if
- If is invertible, then the graph of is the same as the graph of the equation . This is obtained by reflecting the graph of across the line .
- The range of a one-to-one function is the domain of the inverse function , while the domain of is the range of .